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Math Help - Sphere Curve

  1. #1
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    Arrow Sphere Curve

    Please show that the parameter curve : { x=\frac{t}{1+t^2+t^4},y=\frac{t^2}{1+t^2+t^4},z=\f  rac{t^3}{1+t^2+t^4}},( -\infty<t<+\infty) is a sphere curve. then write the formular of the sphere.

    Thanks very much
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  2. #2
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    Quote Originally Posted by Xingyuan View Post
    Please show that the parameter curve : { x=\frac{t}{1+t^2+t^4},y=\frac{t^2}{1+t^2+t^4},z=\f  rac{t^3}{1+t^2+t^4}},( -\infty<t<+\infty) is a sphere curve. then write the formular of the sphere.

    Thanks very much
    let u=1+t^2+t^4. so the curve is: x=\frac{t}{u}, \ y=\frac{t^2}{u}, \ z=\frac{t^3}{u}. we want to find the constants a,b,c,d such that (x-a)^2+(y-b)^2+(z-c)^2=d^2, for all t. i.e.

    (t-au)^2+(t^2-bu)^2+(t^3-cu)^2=d^2u^2. simplifying the LHS gives us: (a^2+b^2+c^2)u^2-(2at+(2b-1)t^2+2ct^3)u=d^2u^2. thus: a=c=0, \ \ b=d=\frac{1}{2}.

    so, just for fun, you can check my result by proving that: \forall t \in \mathbb{R}: \ \ \left(\frac{t}{1+t^2+t^4} \right)^2 + \left(\frac{t^2}{1+t^2+t^4}-\frac{1}{2} \right)^2 + \left(\frac{t^3}{1+t^2+t^4} \right)^2=\frac{1}{4}. \ \ \ \Box
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